Bear in mind the web is a pull environment – not a push one. So whoever is making the data change can not push a notification to anyone else who might be surfing your website at the moment. Instead, it’s every client’s (browser) responsibility to call back to the server ever so often and see what’s happened.
So conceptually, you’ll need to install a javascript timer on the relevant pages (I assume only admin pages, and maybe not even all of them?).
That timer will use ajax to check back with the server every so often. (delay to be determined by you) The ajax call will need to keep track of when it last called the server (timestamp) and send that on every call, so your server can then determine what happened since the client last checked-in.
If the server finds something of interest, it can return a message, or data, to the client and the client’s ajax handler can then display it as a popup.
In your case, it sounds like you’d want to look for posts that were updated since the last client call, presumably of a special post_type, and if you find one or more, then you have something to report about.
Hope this helps a bit!